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I know that delaunay triangulation can be extended into arbitrary dimensions by solving the convex hull problem in $(p+1)$ dimensions and projecting the lower hull into dimension $p$ to obtain a mesh in dimension $p$. However, this approach is asymptotically $\Omega({n}^{{p}/{2}})$. I was wondering if there are any other algorithms to produce arbitrary dimensional meshes that can be implemented faster. If so, what are they, how do they work and what is their asymptotic runtime (in terms of the number of mesh points)?

Paul
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  • What is $\Omega$? What is $d$? – David Ketcheson Jan 15 '12 at 09:00
  • Using asymptotic notation (see Introduction to Algorithms 3rd Edition by Cormen et al.), it roughly means that the runtime for large problem sizes (large point sets) is at least ${n}^{p/2}$, where n is the cardinality of the point set. – Paul Jan 15 '12 at 16:37
  • Thanks. But I found an explanation here that says something different: http://en.wikibooks.org/wiki/Data_Structures/Asymptotic_Notation#Big-Omega_Notation That says that what you wrote only means it is at least $cn^{p/2}$ where $c$ is some constant. Also, since both $n$ and $p$ are arbitrary, can you specify in which one this is asymptotic? – David Ketcheson Jan 15 '12 at 19:15
  • @DavidKetcheson The statement is only a lower bound. The asymptotic statement should be interpreted as being for any given $p$, there is $n_0$ and $c>0$ such that the algorithm takes at least $c n^{p/2}$ for all $n > n_0$. – Jed Brown Jan 15 '12 at 20:00
  • @Paul Do you want a Delaunay mesh? (If the problem is anisotropic, then you would not want Delaunay.) What quality constraints? Note that Delaunay in 3D can still have quite bad elements. – Jed Brown Jan 15 '12 at 20:00
  • Let's say that the mesh is of good enough quality for a finite element simulation (i.e. a mesh that does not produce an ill-conditioned stiffness matrix). – Paul Jan 15 '12 at 20:32
  • @Paul That is not a very well-defined request and the theory is only well-developed for limited cases (e.g. Shewchuck's What Is a Good Linear Finite Element? Interpolation, Conditioning, Anisotropy, and Quality Measures http://www.cs.berkeley.edu/~jrs/papers/elemj.pdf). I don't know of many applications needing finite element computations on arbitrary-dimensional simplicial meshes. I'm afraid you may have to make the question well-posed before someone can reasonably answer it. – Jed Brown Jan 16 '12 at 23:46
  • I see... I wasn't aware that the theory is only well developed for limited cases. I just wanted to know if there is an analogous method to, say, the Marching Cubes algorithm, that applies when meshing a higher dimensional (closed) euclidean region... You're right that there aren't many applications that ring off the top of the head, but that's not to say that there isn't (or never will be). – Paul Jan 17 '12 at 01:12

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How about; Watson, D.F. Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Comput J 24, 167–172 (1981).

http://comjnl.oxfordjournals.org/content/24/2/167.abstract

If I understood it, it should be $\mathcal{O}\left(n^{\frac{2p-1}{n}}\right)$, isn't that better then $Ω\left(n^{p/2}\right)$?

Peter
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